A257617: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = 7*x + 2.

A257617 palindromic γ-positive nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0) · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence from OEIS A257617

\(P_{n}(x) = \left(2 - 5\,x + 7\,n x\right)\,P_{n-1}(x) + \left(7\,x - 7\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	2														4
2	4	36	4													44
3	8	388	388	8												792
4	16	3676	12416	3676	16											19800
5	32	33564	283204	283204	33564	32										633600
6	64	303260	5538184	13027384	5538184	303260	64									24710400
7	128	2732156	99831564	465775352	465775352	99831564	2732156	128								1136678400
8	256	24595932	1717519888	14406600964	27946521120	14406600964	1717519888	24595932	256							60243955200
9	512	221378236	28734710740	406922697244	1371439869268	1371439869268	406922697244	28734710740	221378236	512						3614637312000
10	1024	1992437404	472595309528	10824692284352	59047794756776	101486550325832	59047794756776	10824692284352	472595309528	1992437404	1024					242180699904000
11	2048	17932010364	7691033383708	276378450492720	2323493149205232	6353105331353928	6353105331353928	2323493149205232	276378450492720	7691033383708	17932010364	2048				17921371792896000
12	4096	161388255068	124347638885536	6856621531273580	85734744604734720	353563047869562168	559073269159145664	353563047869562168	85734744604734720	6856621531273580	124347638885536	161388255068	4096			1451631115224576000
13	8192	1452494647868	2002311894318948	166655325219050932	3017722737674824300	18054447958248413976	42630939284350079784	42630939284350079784	18054447958248413976	3017722737674824300	166655325219050932	2002311894318948	1452494647868	8192		127743538139762688000
14	16384	13072452592668	32161904848819816	3991255119689368328	102530865546016396104	864166552404054896612	2922919310089811521104	4348355807003708137968	2922919310089811521104	864166552404054896612	102530865546016396104	3991255119689368328	32161904848819816	13072452592668	16384	12135636123277455360000

Generated b-files: A144827 (row sums), A257617, A257617 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-5b7da9f56b3380cb · revision 1 · JSON
Definition source: oeis_catalog:A257617
Fingerprint: f6-1edf773d06ca8d1b

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
Palindromic rows: yes
γ-positive (gamma_positive): yes
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

OEIS matches ?

A144827: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
A257617: triangle, 36 terms agree, 8 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
A257617: reversed triangle, 36 terms agree, 8 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:   1;
  1:   2,         2;
  2:   4,        36,           4;
  3:   8,       388,         388,            8;
  4:  16,      3676,       12416,         3676,            16;
  5:  32,     33564,      283204,       283204,         33564,            32;
  6:  64,    303260,     5538184,     13027384,       5538184,        303260,           64;
  7: 128,   2732156,    99831564,    465775352,     465775352,      99831564,      2732156,         128;
  8: 256,  24595932,  1717519888,  14406600964,   27946521120,   14406600964,   1717519888,    24595932,       256;
  9: 512, 221378236, 28734710740, 406922697244, 1371439869268, 1371439869268, 406922697244, 28734710740, 221378236, 512;

%F Formula

The row polynomials satisfy P(n,x) = (2+(7*n-5)*x)*P(n-1,x) + (7*x-7*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2 + (7*n - 5)*x)*P[n - 1, x] + (7*x - 7*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2 + (7*n - 5)*x)*v[i - 1] + (7*x - 7*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A257617: triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:   1;
  1:   2,         2;
  2:   4,        36,           4;
  3:   8,       388,         388,            8;
  4:  16,      3676,       12416,         3676,            16;
  5:  32,     33564,      283204,       283204,         33564,            32;
  6:  64,    303260,     5538184,     13027384,       5538184,        303260,           64;
  7: 128,   2732156,    99831564,    465775352,     465775352,      99831564,      2732156,         128;
  8: 256,  24595932,  1717519888,  14406600964,   27946521120,   14406600964,   1717519888,    24595932,       256;
  9: 512, 221378236, 28734710740, 406922697244, 1371439869268, 1371439869268, 406922697244, 28734710740, 221378236, 512;

%F Formula

The row polynomials satisfy P(n,x) = (2+(7*n-5)*x)*P(n-1,x) + (7*x-7*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2 + (7*n - 5)*x)*P[n - 1, x] + (7*x - 7*x^2)*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2 + (7*n - 5)*x)*v[i - 1] + (7*x - 7*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A257617: reversed triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:   1;
  1:   2,         2;
  2:   4,        36,           4;
  3:   8,       388,         388,            8;
  4:  16,      3676,       12416,         3676,            16;
  5:  32,     33564,      283204,       283204,         33564,            32;
  6:  64,    303260,     5538184,     13027384,       5538184,        303260,           64;
  7: 128,   2732156,    99831564,    465775352,     465775352,      99831564,      2732156,         128;
  8: 256,  24595932,  1717519888,  14406600964,   27946521120,   14406600964,   1717519888,    24595932,       256;
  9: 512, 221378236, 28734710740, 406922697244, 1371439869268, 1371439869268, 406922697244, 28734710740, 221378236, 512;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+(7*n-5)*x)*P(n-1,x) + (7*x-7*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2 + (7*n - 5)*x)*P[n - 1, x] + (7*x - 7*x^2)*D[P[n - 1, x], x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2 + (7*n - 5)*x)*v[i - 1] + (7*x - 7*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0)
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-f4ea85fa593cfea8
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-f4ea85fa593cfea8
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-f4ea85fa593cfea8
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-f4ea85fa593cfea8

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

theorem_check liu_wang e-f4ea85fa593cfea8 root: theorem check

Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Used in 3 proof steps

e-f8caac77b40c1335 gamma_polynomial (scale 1, shift 0) → gamma_polynomial of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f…
e-b3016eed90e91248 obreschkoff_combination → obreschkoff_combination of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1…
e-b3016eed90e91248 obreschkoff_combination → obreschkoff_combination of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1…

Related sequences ?

No candidate equalities or related sequences.

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003